Answer :
Let's solve each inequality step-by-step and match each solution on the left side to the correct answer on the right side.
### Inequality 1: [tex]\(-9(x-1) \geq -3(x+5)\)[/tex]
First, we distribute the terms inside the parentheses:
[tex]\[ -9(x-1) \geq -3(x+5) \][/tex]
[tex]\[ -9x + 9 \geq -3x - 15 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -9x + 9 + 3x \geq -3x - 15 + 3x \][/tex]
[tex]\[ -6x + 9 \geq -15 \][/tex]
Subtract 9 from both sides:
[tex]\[ -6x + 9 - 9 \geq -15 - 9 \][/tex]
[tex]\[ -6x \geq -24 \][/tex]
Divide by -6, and remember to reverse the inequality sign:
[tex]\[ x \leq 4 \][/tex]
### Inequality 2: [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex]
Start by eliminating the fraction. Multiply both sides by 5:
[tex]\[ 2x + 7 \leq 5(x-1) \][/tex]
[tex]\[ 2x + 7 \leq 5x - 5 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 2x + 7 - 2x \leq 5x - 5 - 2x \][/tex]
[tex]\[ 7 \leq 3x - 5 \][/tex]
Add 5 to both sides:
[tex]\[ 7 + 5 \leq 3x - 5 + 5 \][/tex]
[tex]\[ 12 \leq 3x \][/tex]
Divide by 3:
[tex]\[ 4 \leq x \][/tex]
Or:
[tex]\[ x \geq 4 \][/tex]
### Inequality 3: [tex]\(2(2x+7) \leq 6(x+2)\)[/tex]
Distribute the constants:
[tex]\[ 2(2x+7) \leq 6(x+2) \][/tex]
[tex]\[ 4x + 14 \leq 6x + 12 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 4x + 14 - 4x \leq 6x + 12 - 4x \][/tex]
[tex]\[ 14 \leq 2x + 12 \][/tex]
Subtract 12 from both sides:
[tex]\[ 14 - 12 \leq 2x + 12 - 12 \][/tex]
[tex]\[ 2 \leq 2x \][/tex]
Divide by 2:
[tex]\[ 1 \leq x \][/tex]
Or:
[tex]\[ x \geq 1 \][/tex]
### Inequality 4: [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex]
Distribute the constants:
[tex]\[ -3(2x-7) \geq -5(x-4) \][/tex]
[tex]\[ -6x + 21 \geq -5x + 20 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -6x + 21 + 5x \geq -5x + 20 + 5x \][/tex]
[tex]\[ -x + 21 \geq 20 \][/tex]
Subtract 21 from both sides:
[tex]\[ -x + 21 - 21 \geq 20 - 21 \][/tex]
[tex]\[ -x \geq -1 \][/tex]
Divide by -1, and remember to reverse the inequality sign:
[tex]\[ x \leq 1 \][/tex]
### Matching the answers:
1. [tex]\(-9(x-1) \geq -3(x+5)\)[/tex] corresponds to [tex]\(x \leq 4\)[/tex].
2. [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex] corresponds to [tex]\(x \geq 4\)[/tex].
3. [tex]\(2(2x+7) \leq 6(x+2)\)[/tex] corresponds to [tex]\(x \geq 1\)[/tex].
4. [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex] corresponds to [tex]\(x \leq 1\)[/tex].
So, the final answers will be:
[tex]\[ \begin{array}{l} -9(x-1) \geq -3(x+5) \implies x \leq 4 \\ \frac{2x+7}{5} \leq x-1 \implies x \geq 4 \\ 2(2x+7) \leq 6(x+2) \implies x \geq 1 \\ -3(2x-7) \geq -5(x-4) \implies x \leq 1 \end{array} \][/tex]
### Inequality 1: [tex]\(-9(x-1) \geq -3(x+5)\)[/tex]
First, we distribute the terms inside the parentheses:
[tex]\[ -9(x-1) \geq -3(x+5) \][/tex]
[tex]\[ -9x + 9 \geq -3x - 15 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -9x + 9 + 3x \geq -3x - 15 + 3x \][/tex]
[tex]\[ -6x + 9 \geq -15 \][/tex]
Subtract 9 from both sides:
[tex]\[ -6x + 9 - 9 \geq -15 - 9 \][/tex]
[tex]\[ -6x \geq -24 \][/tex]
Divide by -6, and remember to reverse the inequality sign:
[tex]\[ x \leq 4 \][/tex]
### Inequality 2: [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex]
Start by eliminating the fraction. Multiply both sides by 5:
[tex]\[ 2x + 7 \leq 5(x-1) \][/tex]
[tex]\[ 2x + 7 \leq 5x - 5 \][/tex]
Next, combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 2x + 7 - 2x \leq 5x - 5 - 2x \][/tex]
[tex]\[ 7 \leq 3x - 5 \][/tex]
Add 5 to both sides:
[tex]\[ 7 + 5 \leq 3x - 5 + 5 \][/tex]
[tex]\[ 12 \leq 3x \][/tex]
Divide by 3:
[tex]\[ 4 \leq x \][/tex]
Or:
[tex]\[ x \geq 4 \][/tex]
### Inequality 3: [tex]\(2(2x+7) \leq 6(x+2)\)[/tex]
Distribute the constants:
[tex]\[ 2(2x+7) \leq 6(x+2) \][/tex]
[tex]\[ 4x + 14 \leq 6x + 12 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ 4x + 14 - 4x \leq 6x + 12 - 4x \][/tex]
[tex]\[ 14 \leq 2x + 12 \][/tex]
Subtract 12 from both sides:
[tex]\[ 14 - 12 \leq 2x + 12 - 12 \][/tex]
[tex]\[ 2 \leq 2x \][/tex]
Divide by 2:
[tex]\[ 1 \leq x \][/tex]
Or:
[tex]\[ x \geq 1 \][/tex]
### Inequality 4: [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex]
Distribute the constants:
[tex]\[ -3(2x-7) \geq -5(x-4) \][/tex]
[tex]\[ -6x + 21 \geq -5x + 20 \][/tex]
Combine like terms to isolate [tex]\(x\)[/tex]:
[tex]\[ -6x + 21 + 5x \geq -5x + 20 + 5x \][/tex]
[tex]\[ -x + 21 \geq 20 \][/tex]
Subtract 21 from both sides:
[tex]\[ -x + 21 - 21 \geq 20 - 21 \][/tex]
[tex]\[ -x \geq -1 \][/tex]
Divide by -1, and remember to reverse the inequality sign:
[tex]\[ x \leq 1 \][/tex]
### Matching the answers:
1. [tex]\(-9(x-1) \geq -3(x+5)\)[/tex] corresponds to [tex]\(x \leq 4\)[/tex].
2. [tex]\(\frac{2x+7}{5} \leq x-1\)[/tex] corresponds to [tex]\(x \geq 4\)[/tex].
3. [tex]\(2(2x+7) \leq 6(x+2)\)[/tex] corresponds to [tex]\(x \geq 1\)[/tex].
4. [tex]\(-3(2x-7) \geq -5(x-4)\)[/tex] corresponds to [tex]\(x \leq 1\)[/tex].
So, the final answers will be:
[tex]\[ \begin{array}{l} -9(x-1) \geq -3(x+5) \implies x \leq 4 \\ \frac{2x+7}{5} \leq x-1 \implies x \geq 4 \\ 2(2x+7) \leq 6(x+2) \implies x \geq 1 \\ -3(2x-7) \geq -5(x-4) \implies x \leq 1 \end{array} \][/tex]
